[Math] Pareto distribution moments

Question

I have 2 questions where I have to use the Pareto distribution.

1. A family of pdf’s that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, $k$ and $y$, both $> 0$, and
   the pdf is:
   $$ f(x;k,\theta)=\frac{k\cdot\theta^k}{x^{k+1}} \ for \ x\geq\theta $$
   and is $ 0 $ otherwise.

a) Sketch the graph of $ f(x;k,\theta) $

b) Verify the the total area under the graph is $ 1 $.

c) For $ \theta<a<b $, obtain an expression for the probability $P(a\leq{X}\leq{b}). $

For this question, I have integrated and proved part b) because all $k $ and $\theta$ cancel. I have also got an expression for part c) by just changing the bounds to $ a $ and $b $. However, I have no idea how to graph the 3 variables, any thoughts?

2. Let X have the Pareto pdf introduced in Exercise 1.

a) If $k>1$, compute $E(X)$

b) What can you say about $E(X)$ if $k=1?$

c) If $k>2$, show that $ V(X)=k\theta^2(k-1)^{-2}(k-2)^{-1} $.

d) If $k=2$, what can you say about $V(C)$?

e) What conditions on $k$ are necessary to make $ E(X^n) $finite?

I got $E(X)=\int_\theta^\infty{x\frac{k\theta^k}{x^{k+1}}}dx=k\theta^k\int_\theta^\infty{x^{-k}}dx=-\frac{k\theta}{-k+1} $. And I got $E(X^2)=\int_\theta^\infty{x^2\frac{k\theta^k}{x^{k+1}}}dx=k\theta^k\int_\theta^\infty{x^{-k+1}dx}=\frac{-k\theta^2}{-k+2}$

Answer 1

In light of the fact that $$\frac{1}{x^\alpha}\to 0\text{ as }x\to\infty\quad\text{ provided } \alpha>0$$

, we have the $r$-th order raw moment of $X$ about $0$ :

\begin{align} E(X^r)&=\int_{\theta}^\infty \frac{x^r\,k\theta^k}{x^{k+1}}\,dx \\&=k\theta^k\int_{\theta}^\infty x^{r-k-1}\,dx \\&=k\theta^k\lim_{A\to\infty}\left[\frac{x^{-(k-r)}}{-(k-r)}\right]_{\theta}^A \\&=\frac{k\theta^r}{k-r}\qquad,\text{ if }k>r \end{align}

You should be able to proceed now.